A Distance of the touch-point and the midpoint ?

[lvl107]

Hello Everyone.
Let ABC be a triangle with Incircle ( I ), and let ( I ) touch BC, CA, AB at A', B', C', respectively, and A'',
B'', C'' are the midpoints of the sides BC, CA, AB, respectively.
Prove :
One of the distances A'A", B'B", C'C" is equal of the sum of the two others. ( C'C'' + A'A" = B'B'' )

Thanks in advance for your time and help.
Best Regards.
Loi.